# How do you solve sqrt(5x^2-9)=2x and check your solution?

Sep 4, 2017

$x = + 3$
(see below for solution and check)

#### Explanation:

Given
$\textcolor{w h i t e}{\text{XXX}} \sqrt{5 {x}^{2} - 9} = 2 x$

Squaring both sides (this may introduce extraneous solutions; we will need to check later)
$\textcolor{w h i t e}{\text{XXX}} 5 {x}^{2} - 9 = 4 {x}^{2}$

Subtract $4 {x}^{2}$ from both sides
$\textcolor{w h i t e}{\text{XXX}} {x}^{2} - 9 = 0$

Factor the left side
$\textcolor{w h i t e}{\text{XXX}} \left(x + 3\right) \left(x - 3\right) = 0$

which implies
$\textcolor{w h i t e}{\text{XXX")x=-3color(white)("xxx") or color(white)("xxx}} x = + 3$

Checking for extraneous solutions:
{: ("with "x=-3,,color(white)("xxx"),"with "x=+3,), (sqrt(5 * x^2-9)=+6,2x=-6,,sqrt(5^2-9)=+6,2x=6), ("extraneous",,,"valid",) :}

The only valid solution is $x = + 3$